Question

I am trying to solve a simple linear equations system using LAPACK. I use dbsvg method which is optimised for banded matrices. I’ve obsereved a realy strange behaviour. When I fill the AT matrix this way:

for(i=0; i<DIM;i++) AB[0][i] = -1;
for(i=0; i<DIM;i++) AB[1][i] = 2;
for(i=0; i<DIM;i++) AB[2][i] = -1;
for(i=0; i<3; i++)
    for(j=0;j<DIM;j++) {
        AT[i*DIM+j]=AB[i][j];
    }

And call:

dgbsv_(&N, &KL, &KU, &NRHS, AT, &LDAB, myIpiv, x, &LDB, &INFO);

It works perfectly. However, when I do it this way:

for(i=0; i<DIM;i++) AT[i] = -1;
for(i=0; i<DIM;i++) AT[DIM+i] = 2;
for(i=0; i<DIM;i++) AT[2*DIM+i] = -1;

It results with a vector filled with NaN. Here are the declarations:

double AB[3][DIM], AT[3*DIM];
double x[DIM];
int myIpiv[DIM];
int N=DIM, KL=1, KU=1, NRHS=1, LDAB=DIM, LDB=DIM, INFO;

Any ideas?

Answer 1

You’re not laying out the entries in the band storage properly; it was working before by a happy accident. The LAPACK docs say:

    On entry, the matrix A in band storage, in rows KL+1 to
    2*KL+KU+1; rows 1 to KL of the array need not be set.
    The j-th column of A is stored in the j-th column of the
    array AB as follows:
    AB(KL+KU+1+i-j,j) = A(i,j) for max(1,j-KU)<=i<=min(N,j+KL)
    On exit, details of the factorization: U is stored as an
    upper triangular band matrix with KL+KU superdiagonals in
    rows 1 to KL+KU+1, and the multipliers used during the
    factorization are stored in rows KL+KU+2 to 2*KL+KU+1.
    See below for further details.

So if you want a tridiagonal matrix with 2 on the diagonal and -1 above and below, the layout should be:

 *  *  *  *  *  *  *  ...  *  *  *  *
 * -1 -1 -1 -1 -1 -1  ... -1 -1 -1 -1
 2  2  2  2  2  2  2  ...  2  2  2  2
-1 -1 -1 -1 -1 -1 -1  ... -1 -1 -1  *

LDAB should be 4 in this case. Bear in mind that LAPACK uses a column-major layout, so the actual array should be look like this in memory:

{ *, *, 2.0, -1.0, *, -1.0, 2.0, -1.0, *, -1.0, 2.0, -1.0, ... }

dgbsv was giving different results for the two identical arrays because it was reading off the ends of the arrays that you had laid out.